61.4.32 problem Problem 5(c)

Internal problem ID [15357]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 5(c)
Date solved : Thursday, October 02, 2025 at 10:12:03 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+29 y&=5 \delta \left (t -\pi \right )-5 \delta \left (t -2 \pi \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.293 (sec). Leaf size: 36

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+29*y(t) = 5*Dirac(t-Pi)-5*Dirac(t-2*Pi); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\left (\operatorname {Heaviside}\left (t -2 \pi \right ) {\mathrm e}^{2 \pi }+\operatorname {Heaviside}\left (t -\pi \right )\right ) \sin \left (5 t \right ) {\mathrm e}^{2 \pi -2 t} \]

✓ Mathematica. Time used: 0.075 (sec). Leaf size: 93

ode=D[y[t],{t,2}]+4*D[y[t],t]+29*y[t]==5*DiracDelta[t-Pi]-5*DiracDelta[t-2*Pi]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -e^{-2 t} \sin (5 t) \left (\int _1^0-e^{2 K[1]} \cos (5 K[1]) (\delta (K[1]-2 \pi )-\delta (K[1]-\pi ))dK[1]-\int _1^t-e^{2 K[1]} \cos (5 K[1]) (\delta (K[1]-2 \pi )-\delta (K[1]-\pi ))dK[1]\right ) \end{align*}

✓ Sympy. Time used: 5.161 (sec). Leaf size: 160

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*Dirac(t - 2*pi) - 5*Dirac(t - pi) + 29*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (\int \operatorname {Dirac}{\left (t - 2 \pi \right )} e^{2 t} \sin {\left (5 t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \pi \right )} e^{2 t} \sin {\left (5 t \right )}\, dt - \int \operatorname {Dirac}{\left (t - \pi \right )} e^{2 t} \sin {\left (5 t \right )}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - \pi \right )} e^{2 t} \sin {\left (5 t \right )}\, dt\right ) \cos {\left (5 t \right )} + \left (- \int \operatorname {Dirac}{\left (t - 2 \pi \right )} e^{2 t} \cos {\left (5 t \right )}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \pi \right )} e^{2 t} \cos {\left (5 t \right )}\, dt + \int \operatorname {Dirac}{\left (t - \pi \right )} e^{2 t} \cos {\left (5 t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - \pi \right )} e^{2 t} \cos {\left (5 t \right )}\, dt\right ) \sin {\left (5 t \right )}\right ) e^{- 2 t} \]